We consider closed-form approximations for European put option prices within the Heston and GARCH diffusion stochastic volatility models with time-dependent parameters. Our methodology involves writing the put option price as an expectation of a Black-Scholes formula and performing a second-order Taylor expansion around the mean of its argument. The difficulties then faced are simplifying a number of expectations induced by the Taylor expansion. Under the assumption of piecewise-constant parameters, we derive closed-form pricing formulas and devise a fast calibration scheme. Furthermore, we perform a numerical error and sensitivity analysis to investigate the quality of our approximation and show that the errors are well within the acceptable range for application purposes. Lastly, we derive bounds on the remainder term generated by the Taylor expansion.

我们在 Heston 和 GARCH 扩散随机波动率模型中考虑了欧式看跌期权价格的封闭式近似，该模型具有时间相关参数。我们的方法包括将看跌期权价格写成 Black-Scholes 公式的期望值，并围绕其参数的均值执行二阶泰勒展开。然后面临的困难是简化由泰勒展开引起的一些期望。在分段常数参数的假设下，我们推导了封闭式定价公式并设计了一种快速校准方案。此外，我们进行了数值误差和灵敏度分析，以研究我们的近似值的质量，并表明误差在应用目的可接受的范围内。最后，